Symbols

This post contains various definitions of the symbol of a differential operator. We will state a local version, then a global version and then we will finally view the symbol in its most abstract form: a section of a bundle over the total space of a cotangent bundle.

Review of Local Definitions

Let’s start by recalling that a differential operator of order on the manifold is is defined by:

where is smooth and if , then

Let and . The Symbol of , denoted by is then

A differential operator is said to be elliptic if for all and every we have that is invertible.

Global definition of the Symbol

Consider a globally defined differential operator

for and we want to define a linear map
in a coordinate free way.

With this in mind let and choose:
1. such that
2. such that

Then we define

Notice that even though this is a coordinate free definition of the symbol, it is still unclear how it changes in and . We will later see that is actually smooth on . Before this, we should prove that this definition is in fact independent on the choice of and .

does not depend on

Claim 1 If is a smooth function such that and , then

Proof: For any differential operator , any section and any function ,

Setting we have

Induction on the order of and (3) will give us the result:

Let , then by definition

and so

Now assume the claim is true for every differential operator of order less than and suppose . By definition,
Thus, by induction

and notice that (3) gives us

so that

does not depend on

Claim 2 Let be such that , then

Proof: It is easier if we use the easy direction of Peetre’s Theorem so that we can use the fact that is local, that is

equivalently

equivalently

So, since , we have as sought.

Let us finish the section with a short remark:

is homogeneous of degree in . That is, for every ,

Proof: Simply take instead of in the definition for .

Local=Global

Lemma 1 For a differential operator of order , the two definitions of symbol coincide under the identification given by

Proof: Let . The function satisfies the conditions stated in the coordinate free definition of . Let be the constant section , that is, for every .

Then (2) reads

where by (1)

Notice that here

since is a constant section.

Also notice that
1. for every : This is because there is always a factor of in the expression for whenever .
2. : This is a simple calculation.

Consolidating all the information we conclude

Symbol as a section

By consolidating definitions (*) and (1) of we get . Here is the bundle map and we are just looking at the diagram

To be explicit, if , then with . So

that is, and we are using the identification .

Smoothness follows from the smoothness of the local definition and the fact that both definitions coincide locally.

Finally, let

then we have

Proposition 2 There is an exact sequence

Notice that this proposition (re)captures the fact that the symbol of an operator only `sees’ the `top’ degree of the operator.

Fundamental Theorem of Elliptic Operators

Now that we have a global definition of the symbol of a differential operator, we can state what it means for a differential operator to be elliptic. Namely, is elliptic if for every (i.e is in the complement of the zero section of the cotangent bundle), the map is invertible. The most important result involving elliptic operators is the following theorem:

Theorem 3 Fundamental Theorem of Elliptic Operators
If is an elliptic differential operator over a compact manifold , then both and are finite dimensional vector spaces.